While doing an inequality, I encountered the following expression,where $ABC$ is a triangle: $$\cos \left(\frac{B-C}{2}\right) + \cos \left(\frac{C-A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$$
So my question is do you know some quantites related to that expression (or even the individual terms or something related) in a triangle?
It could be better if its in terms of $R$ and $r$ :)
The question is not clear enough to give a complete answer but I can give you a hint that might help. The altitude and diameter drawn from $A$ form angle of measure $\angle B-\angle C$ whose bisector coincides with the bisector of $\angle BAC$. Now let $M$ be the intersection of the bisector of $\angle BAC$ and the circumcircle. It is easy to verify that $\cos(\frac{\angle B -\angle C}{2})=\frac {AM}{2R}$. You can obtain similar terms for vertices $B$ and $C$ as well. Now see if you can express $AM$ (and the analogous terms for $B$ and $C$) in a way you'd like to have.